Data 621 - Homework 1

Data 621 Homework1

Introduction

A wealth of statistics are collected in sports, and baseball is no exception. The exploration and modeling that follows is based on a “Moneyball” dataset where the response variable is the number of wins for a given team for a particular season.

data_test <- read.csv("https://raw.githubusercontent.com/salma71/Data_621/master/HW_1/datasets/moneyball-evaluation-data.csv", header = TRUE) %>%select(-INDEX)
data_train <- read.csv("https://raw.githubusercontent.com/salma71/Data_621/master/HW_1/datasets/moneyball-training-data.csv", header = TRUE) %>% select(-INDEX)

Data Exploration

Data Exploration

The dataset is composed of 2276 observations ranging from 1871 to 2006. Sixteen variables were used to record the various pitching, batting and fielding efforts for the teams. These are listed below.

dim(data_train)

[1] 2276   16

MI: do we even need to describe the test set?

dim(data_test)

[1] 259  15

# list types for each attribute
sapply(data_train, class)

     TARGET_WINS   TEAM_BATTING_H  TEAM_BATTING_2B  TEAM_BATTING_3B 
       "integer"        "integer"        "integer"        "integer" 
 TEAM_BATTING_HR  TEAM_BATTING_BB  TEAM_BATTING_SO  TEAM_BASERUN_SB 
       "integer"        "integer"        "integer"        "integer" 
 TEAM_BASERUN_CS TEAM_BATTING_HBP  TEAM_PITCHING_H TEAM_PITCHING_HR 
       "integer"        "integer"        "integer"        "integer" 
TEAM_PITCHING_BB TEAM_PITCHING_SO  TEAM_FIELDING_E TEAM_FIELDING_DP 
       "integer"        "integer"        "integer"        "integer" 

skim(data_train)

Data summary

Name	data_train
Number of rows	2276
Number of columns	16
_______________________
Column type frequency:
numeric	16
________________________
Group variables	None

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
TARGET_WINS	0	1.00	80.79	15.75	0	71.0	82.0	92.00	146	▁▁▇▅▁
TEAM_BATTING_H	0	1.00	1469.27	144.59	891	1383.0	1454.0	1537.25	2554	▁▇▂▁▁
TEAM_BATTING_2B	0	1.00	241.25	46.80	69	208.0	238.0	273.00	458	▁▆▇▂▁
TEAM_BATTING_3B	0	1.00	55.25	27.94	0	34.0	47.0	72.00	223	▇▇▂▁▁
TEAM_BATTING_HR	0	1.00	99.61	60.55	0	42.0	102.0	147.00	264	▇▆▇▅▁
TEAM_BATTING_BB	0	1.00	501.56	122.67	0	451.0	512.0	580.00	878	▁▁▇▇▁
TEAM_BATTING_SO	102	0.96	735.61	248.53	0	548.0	750.0	930.00	1399	▁▆▇▇▁
TEAM_BASERUN_SB	131	0.94	124.76	87.79	0	66.0	101.0	156.00	697	▇▃▁▁▁
TEAM_BASERUN_CS	772	0.66	52.80	22.96	0	38.0	49.0	62.00	201	▃▇▁▁▁
TEAM_BATTING_HBP	2085	0.08	59.36	12.97	29	50.5	58.0	67.00	95	▂▇▇▅▁
TEAM_PITCHING_H	0	1.00	1779.21	1406.84	1137	1419.0	1518.0	1682.50	30132	▇▁▁▁▁
TEAM_PITCHING_HR	0	1.00	105.70	61.30	0	50.0	107.0	150.00	343	▇▇▆▁▁
TEAM_PITCHING_BB	0	1.00	553.01	166.36	0	476.0	536.5	611.00	3645	▇▁▁▁▁
TEAM_PITCHING_SO	102	0.96	817.73	553.09	0	615.0	813.5	968.00	19278	▇▁▁▁▁
TEAM_FIELDING_E	0	1.00	246.48	227.77	65	127.0	159.0	249.25	1898	▇▁▁▁▁
TEAM_FIELDING_DP	286	0.87	146.39	26.23	52	131.0	149.0	164.00	228	▁▂▇▆▁

skim(data_test)

Data summary

Name	data_test
Number of rows	259
Number of columns	15
_______________________
Column type frequency:
numeric	15
________________________
Group variables	None

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
TEAM_BATTING_H	0	1.00	1469.39	150.66	819	1387.0	1455.0	1548.00	2170	▁▂▇▁▁
TEAM_BATTING_2B	0	1.00	241.32	49.52	44	210.0	239.0	278.50	376	▁▂▇▇▂
TEAM_BATTING_3B	0	1.00	55.91	27.14	14	35.0	52.0	72.00	155	▇▇▃▁▁
TEAM_BATTING_HR	0	1.00	95.63	56.33	0	44.5	101.0	135.50	242	▆▅▇▃▁
TEAM_BATTING_BB	0	1.00	498.96	120.59	15	436.5	509.0	565.50	792	▁▁▅▇▁
TEAM_BATTING_SO	18	0.93	709.34	243.11	0	545.0	686.0	912.00	1268	▁▃▇▇▂
TEAM_BASERUN_SB	13	0.95	123.70	93.39	0	59.0	92.0	151.75	580	▇▃▁▁▁
TEAM_BASERUN_CS	87	0.66	52.32	23.10	0	38.0	49.5	63.00	154	▂▇▃▁▁
TEAM_BATTING_HBP	240	0.07	62.37	12.71	42	53.5	62.0	67.50	96	▃▇▅▁▁
TEAM_PITCHING_H	0	1.00	1813.46	1662.91	1155	1426.5	1515.0	1681.00	22768	▇▁▁▁▁
TEAM_PITCHING_HR	0	1.00	102.15	57.65	0	52.0	104.0	142.50	336	▇▇▆▁▁
TEAM_PITCHING_BB	0	1.00	552.42	172.95	136	471.0	526.0	606.50	2008	▆▇▁▁▁
TEAM_PITCHING_SO	18	0.93	799.67	634.31	0	613.0	745.0	938.00	9963	▇▁▁▁▁
TEAM_FIELDING_E	0	1.00	249.75	230.90	73	131.0	163.0	252.00	1568	▇▁▁▁▁
TEAM_FIELDING_DP	31	0.88	146.06	25.88	69	131.0	148.0	164.00	204	▁▂▇▇▂

---

Missing Data

The dataset, while mostly complete, is missing a significant number of observations for some of the varibales, namely batters hit by pitchers (HBP) and runners caught stealing bases (CS). We recognize that these are relatively rare occurences in typical baseball games. Nevertheless, the treatment of the missing data depends on the particular model being evaluated. Some models ignore these variables altgother, while other use these variables as a part of engineered features.

data_train %>% 
  gather(variable, value) %>%
  filter(is.na(value)) %>%
  group_by(variable) %>%
  tally() %>%
  mutate(percent = n / nrow(data_train) * 100) %>%
  mutate(percent = paste0(round(percent, ifelse(percent < 10, 1, 0)), "%")) %>%
  arrange(desc(n)) %>%
  rename(`Variable Missing Data` = variable,
         `Number of Records` = n,
         `Share of Total` = percent) %>%
  kable() %>%
  kable_styling()

Variable Missing Data	Number of Records	Share of Total
TEAM_BATTING_HBP	2085	92%
TEAM_BASERUN_CS	772	34%
TEAM_FIELDING_DP	286	13%
TEAM_BASERUN_SB	131	5.8%
TEAM_BATTING_SO	102	4.5%
TEAM_PITCHING_SO	102	4.5%

data_test %>% 
  gather(variable, value) %>%
  filter(is.na(value)) %>%
  group_by(variable) %>%
  tally() %>%
  mutate(percent = n / nrow(data_train) * 100) %>%
  mutate(percent = paste0(round(percent, ifelse(percent < 10, 1, 0)), "%")) %>%
  arrange(desc(n)) %>%
  rename(`Variable Missing Data` = variable,
         `Number of Records` = n,
         `Share of Total` = percent) %>%
  kable() %>%
  kable_styling()

Variable Missing Data	Number of Records	Share of Total
TEAM_BATTING_HBP	240	11%
TEAM_BASERUN_CS	87	3.8%
TEAM_FIELDING_DP	31	1.4%
TEAM_BATTING_SO	18	0.8%
TEAM_PITCHING_SO	18	0.8%
TEAM_BASERUN_SB	13	0.6%

---

Visualization

Visualizing the data reveals a variety of distributions. Some variables are approximiately normal, while other are bimodal or skewed and in some cases extremely skewed especially the variable related to pitching. This skew suggests special treatment of some variables in order to respect the underlying normality assumptions of the models that follow.

data_train %>% 
  gather() %>% 
  ggplot(aes(x= value)) + 
  geom_density(fill='pink') + 
  facet_wrap(~key, scales = 'free')

MI: I think we should remove the SO histogram below. The TARGET_WINS plot is also already shown above. Or maybe only give a blow up plot of the response variable.

Let’s take a closer look at the TEAM_BASERUN_SB

data_train %>% 
  ggplot(aes(TEAM_BASERUN_SB)) + 
  geom_histogram(bins = 50, fill = 'pink') +
  geom_vline(aes(xintercept = mean(TEAM_BASERUN_SB, na.rm = T)), col = "red", lty = 2) +
  geom_vline(aes(xintercept = median(TEAM_BASERUN_SB, na.rm = T)), col = "blue", lty = 2) +
  labs(x = element_blank(),
       y = "Count",
       title = "Distribution of Stolen Bases",
       caption = "* Red line is the mean value and blue is the median") + 
  theme_classic()

ggplot(data_train, aes(x=TARGET_WINS)) + 
    geom_histogram(aes(y=..density..),binwidth = 3, colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic()

Correlations with Response Variable

Let’s take a look at how the predictors are correlated with the response variable:

A visualization of correlation between the variables reveals both expected an unexpected observations: - Most puzzling is the very high correlation between the number of homeruns pitched and the number of homeruns batted. These variables are expected to be unrelated as one variable represents an advantage for the batting team, while the other is an advantage for the opposing team. - The response variable “TARGET_WINS” is most highly correlated with “TEAM_BATTING_H” which is sensible as this represnts the number of base hits by batters. More hits would suggest more opportunities to run around the bases and make it home to score points.

q <- cor(data_train)
ggcorrplot(q, type = "upper", outline.color = "white",
           ggtheme = theme_classic,
           colors = c("#6D9EC1", "white", "#E46726"),
           lab = TRUE, show.legend = FALSE, tl.cex = 8, lab_size = 3) 

A scatterplot with simple linear regression lines displays the relationships along with the distribution of the data. These distributions provide indications of non-normality as well as the influence of outliers. These will be dealt with via variable transformation and deletion of individuals observations determined to be biasing the model. (MI: explore this a bit more)

data_train %>%
  gather(variable, value, -TARGET_WINS) %>%
  ggplot(., aes(value, TARGET_WINS)) + 
  geom_point(fill = "pink", color="pink") + 
  geom_smooth(method = "lm", se = FALSE, color = "red") + 
  facet_wrap(~variable, scales ="free", ncol = 4) +
  labs(x = element_blank(), y = "Wins")

---

Data Preparation

Imputation using KNN

As mentioned earlier, some variable have incomplete data. The following imputation schemes aim to fill in these data voids.

sum(is.na(data_train))/prod(dim(data_train))

[1] 0.09550747

data_train <- data_train %>%
  mutate(TEAM_BATTING_SO = ifelse(TEAM_BATTING_SO == 0, NA, TEAM_BATTING_SO)) %>%
  mutate(TEAM_PITCHING_SO = ifelse(TEAM_PITCHING_SO > 5346, NA, TEAM_PITCHING_SO)) %>%
  select(-TEAM_BATTING_HBP)

set.seed(42)
knn <- data_train %>% knnImputation()
impute_me <- is.na(data_train$TEAM_BATTING_SO)
data_train[impute_me,"TEAM_BATTING_SO"] <- knn[impute_me,"TEAM_BATTING_SO"] 
impute_me <- is.na(data_train$TEAM_BASERUN_SB)
data_train[impute_me,"TEAM_BASERUN_SB"] <- knn[impute_me,"TEAM_BASERUN_SB"] 
impute_me <- is.na(data_train$TEAM_BASERUN_CS)
data_train[impute_me,"TEAM_BASERUN_CS"] <- knn[impute_me,"TEAM_BASERUN_CS"] 
impute_me <- is.na(data_train$TEAM_PITCHING_SO)
data_train[impute_me,"TEAM_PITCHING_SO"] <- knn[impute_me,"TEAM_PITCHING_SO"]
impute_me <- is.na(data_train$TEAM_FIELDING_DP)
data_train[impute_me,"TEAM_FIELDING_DP"] <- knn[impute_me,"TEAM_FIELDING_DP"]

sum(is.na(data_train))/prod(dim(data_train))

[1] 0

---

Do the same for the data_test

sum(is.na(data_test))/prod(dim(data_test))

[1] 0.1047619

data_test <- data_test %>%
  mutate(TEAM_BATTING_SO = ifelse(TEAM_BATTING_SO == 0, NA, TEAM_BATTING_SO)) %>%
  mutate(TEAM_PITCHING_SO = ifelse(TEAM_PITCHING_SO > 5346, NA, TEAM_PITCHING_SO)) %>%
  select(-TEAM_BATTING_HBP)

set.seed(42)
knn <- data_test %>% knnImputation()
impute_me <- is.na(data_test$TEAM_BATTING_SO)
data_test[impute_me,"TEAM_BATTING_SO"] <- knn[impute_me,"TEAM_BATTING_SO"] 
impute_me <- is.na(data_test$TEAM_BASERUN_SB)
data_test[impute_me,"TEAM_BASERUN_SB"] <- knn[impute_me,"TEAM_BASERUN_SB"] 
impute_me <- is.na(data_test$TEAM_BASERUN_CS)
data_test[impute_me,"TEAM_BASERUN_CS"] <- knn[impute_me,"TEAM_BASERUN_CS"] 
impute_me <- is.na(data_test$TEAM_PITCHING_SO)
data_test[impute_me,"TEAM_PITCHING_SO"] <- knn[impute_me,"TEAM_PITCHING_SO"]
impute_me <- is.na(data_test$TEAM_FIELDING_DP)
data_test[impute_me,"TEAM_FIELDING_DP"] <- knn[impute_me,"TEAM_FIELDING_DP"]

sum(is.na(data_test))/prod(dim(data_test))

[1] 0

MI: this passage is just for reference when dealing with leverage point. Can be removed later

Outliers & Leverage Points

In summary, an outlier is a point whose standardized residual falls outside the interval from –2 to 2. Recall that a bad leverage point is a leverage point which is also an outlier. Thus, a bad leverage point is a leverage point whose standar- dized residual falls outside the interval from –2 to 2. On the other hand, a good leverage point is a leverage point whose standardized residual falls inside the interval from –2 to 2.

Recall that the rule for simple linear regression for classifying a point as a leverage point is hii > 4/n .

---

Data Transformation

As seen on the scatterplots above, the spread of the data in some of the variables suggests that transformations might help in normalizing the variability of the data. Log transformations are used here for that purpose as seen on the comparative histograms below.

# New variable: TEAM_BATTING_1B
temp_data <- read.csv("https://raw.githubusercontent.com/salma71/Data_621/master/HW_1/datasets/moneyball-training-data.csv", header = TRUE) %>% select(-INDEX)
base_data <- temp_data
mod_data <- base_data %>% mutate(TEAM_BATTING_1B = base_data$TEAM_BATTING_H - select(., TEAM_BATTING_2B:TEAM_BATTING_HR) %>% rowSums(na.rm = FALSE))
head(mod_data)

data_transformed <- mod_data

#Log transform TEAM_BASERUN_CS
data_transformed$TEAM_BASERUN_CS_tform <-log(data_transformed$TEAM_BASERUN_CS)
baserun_cs <- ggplot(data_transformed, aes(x=TEAM_BASERUN_CS)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BASERUN_CS")
baserun_cs_tf <- ggplot(data_transformed, aes(x=TEAM_BASERUN_CS_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "Log Transformed")

#Log transform TEAM_BASERUN_SB
data_transformed$TEAM_BASERUN_SB_tform <-log(data_transformed$TEAM_BASERUN_SB)
baserun_sb <- ggplot(data_transformed, aes(x=TEAM_BASERUN_SB)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BASERUN_SB")
baserun_sb_tf <- ggplot(data_transformed, aes(x=TEAM_BASERUN_SB_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "Log Transformed")

#Log transform TEAM_BATTING_3B
data_transformed$TEAM_BATTING_3B_tform <-log(data_transformed$TEAM_BATTING_3B)
batting_3b <- ggplot(data_transformed, aes(x=TEAM_BATTING_3B)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BATTING_3B")
batting_3b_tf <- ggplot(data_transformed, aes(x=TEAM_BATTING_3B_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "Log Transformed")

#BoxCoxtransform TEAM_BATTING_BB
data_transformed$TEAM_BATTING_BB_tform <- BoxCox(data_transformed$TEAM_BATTING_BB, BoxCoxLambda(data_transformed$TEAM_BATTING_BB))
batting_bb <- ggplot(data_transformed, aes(x=TEAM_BATTING_BB)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BATTING_BB")
batting_bb_tf <- ggplot(data_transformed, aes(x=TEAM_BATTING_BB_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "BoxCox Transformed")

#BoxCoxtransform TEAM_BATTING_H
data_transformed$TEAM_BATTING_H_tform <- BoxCox(data_transformed$TEAM_BATTING_H, BoxCoxLambda(data_transformed$TEAM_BATTING_H))
batting_h <- ggplot(data_transformed, aes(x=TEAM_BATTING_H)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BATTING_H")
batting_h_tf <- ggplot(data_transformed, aes(x=TEAM_BATTING_H_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "BoxCox Transformed")

#BoxCoxtransform TEAM_BATTING_1B
data_transformed$TEAM_BATTING_1B_tform <- BoxCox(data_transformed$TEAM_BATTING_1B, BoxCoxLambda(data_transformed$TEAM_BATTING_1B))
batting_1b <- ggplot(data_transformed, aes(x=TEAM_BATTING_1B)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_BATTING_1B")
batting_1b_tf <- ggplot(data_transformed, aes(x=TEAM_BATTING_1B_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "BoxCox Transformed")

#BoxCoxtransform TEAM_FIELDING_E
data_transformed$TEAM_FIELDING_E_tform <- BoxCox(data_transformed$TEAM_FIELDING_E, BoxCoxLambda(data_transformed$TEAM_FIELDING_E))
fielding_e <- ggplot(data_transformed, aes(x=TEAM_FIELDING_E)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_FIELDING_E")
fielding_e_tf <- ggplot(data_transformed, aes(x=TEAM_FIELDING_E_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "BoxCox Transformed")

#Log transform TEAM_PITCHING_BB
data_transformed$TEAM_PITCHING_BB_tform <-log(data_transformed$TEAM_PITCHING_BB)
pitching_bb <- ggplot(data_transformed, aes(x=TEAM_PITCHING_BB)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_PITCHING_BB")
pitching_bb_tf <- ggplot(data_transformed, aes(x=TEAM_PITCHING_BB_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "Log Transformed")

#BoxCoxtransform TEAM_PITCHING_H
data_transformed$TEAM_PITCHING_H_tform <- BoxCox(data_transformed$TEAM_PITCHING_H, BoxCoxLambda(data_transformed$TEAM_PITCHING_H))
pitching_h <- ggplot(data_transformed, aes(x=TEAM_PITCHING_H)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_PITCHING_H")
pitching_h_tf <- ggplot(data_transformed, aes(x=TEAM_PITCHING_H_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "BoxCox Transformed")

#Log transform TEAM_PITCHING_SO
data_transformed$TEAM_PITCHING_SO_tform <-log(data_transformed$TEAM_PITCHING_SO)
pitching_so <- ggplot(data_transformed, aes(x=TEAM_PITCHING_SO)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "TEAM_PITCHING_SO")
pitching_so_tf <- ggplot(data_transformed, aes(x=TEAM_PITCHING_SO_tform)) + 
    geom_histogram(aes(y=..density..), colour="black", fill="red") +
    geom_density(alpha=.8, fill="pink") + 
  theme_classic() + labs(title = "Log Transformed")
plot_grid(baserun_cs, baserun_cs_tf, baserun_sb, baserun_sb_tf,
          batting_3b, batting_3b_tf, batting_bb, batting_bb_tf,
          batting_h, batting_h_tf, batting_1b, batting_1b_tf, 
          fielding_e, fielding_e_tf, pitching_bb, pitching_bb_tf, 
          pitching_h, pitching_h_tf, pitching_so, pitching_so_tf, 
          ncol = 2)

Do the same for the test set

#Test data transformations to match model
temp_test <- read.csv("https://raw.githubusercontent.com/salma71/Data_621/master/HW_1/datasets/moneyball-evaluation-data.csv", header = TRUE) %>%select(-INDEX)

mod_data_test <- temp_test %>% mutate(TEAM_BATTING_1B = TEAM_BATTING_H - select(., TEAM_BATTING_2B:TEAM_BATTING_HR) %>% rowSums(na.rm = FALSE))

test_data_transformed <- mod_data_test

#Log transform TEAM_BASERUN_CS
test_data_transformed$TEAM_BASERUN_CS_tform <-log(test_data_transformed$TEAM_BASERUN_CS)

#Log transform TEAM_BASERUN_SB
test_data_transformed$TEAM_BASERUN_SB_tform <-log(test_data_transformed$TEAM_BASERUN_SB)

#Log transform TEAM_BATTING_3B
test_data_transformed$TEAM_BATTING_3B_tform <-log(test_data_transformed$TEAM_BATTING_3B)

#BoxCoxtransform TEAM_BATTING_BB
test_data_transformed$TEAM_BATTING_BB_tform <- BoxCox(test_data_transformed$TEAM_BATTING_BB, BoxCoxLambda(test_data_transformed$TEAM_BATTING_BB))

#BoxCoxtransform TEAM_BATTING_H
test_data_transformed$TEAM_BATTING_H_tform <- BoxCox(test_data_transformed$TEAM_BATTING_H, BoxCoxLambda(test_data_transformed$TEAM_BATTING_H))

#BoxCoxtransform TEAM_BATTING_1B
test_data_transformed$TEAM_BATTING_1B_tform <- BoxCox(test_data_transformed$TEAM_BATTING_1B, BoxCoxLambda(test_data_transformed$TEAM_BATTING_1B))

#BoxCoxtransform TEAM_FIELDING_E
test_data_transformed$TEAM_FIELDING_E_tform <- BoxCox(test_data_transformed$TEAM_FIELDING_E, BoxCoxLambda(test_data_transformed$TEAM_FIELDING_E))

#Log transform TEAM_PITCHING_BB
test_data_transformed$TEAM_PITCHING_BB_tform <-log(test_data_transformed$TEAM_PITCHING_BB)

#BoxCoxtransform TEAM_PITCHING_H
test_data_transformed$TEAM_PITCHING_H_tform <- BoxCox(test_data_transformed$TEAM_PITCHING_H, BoxCoxLambda(test_data_transformed$TEAM_PITCHING_H))

#Log transform TEAM_PITCHING_SO
test_data_transformed$TEAM_PITCHING_SO_tform <-log(test_data_transformed$TEAM_PITCHING_SO)

---

Feature Engineering

Research into baseball statistics suggests the use of the following engineered variables which are composites of variables from the base dataset. These variables, namely “at bats”, “batting average”, “on base percentage” and "slugging percentage’ provide more insight into a team’s batting performance by providing variables quantifying the number of opportunities of hitting the ball, the number of times the ball was actually hit, and when hit, how many bases the batter was able to reach. All these variables are representations of a team’s ability to score points. (Maybe discuss variables that we expect to benefit the opposing team).

# Creating "singles by batters"
# Creating "at bats" variable representing every time a batter steps up to bat
# Creating "batting average" variable
# Creating "on base percentage" representing the proportion of ways to get a base out of total opportunities to hit the ball
# Creating "slugging percentage" which is a weighted sum of hits by number of bases acquired divided by opportunities to hit the ball
add_advanced_bb_features <- function(df) {
  df %>%
    mutate(TEAM_BATTING_1B = TEAM_BATTING_H - TEAM_BATTING_2B - TEAM_BATTING_3B - TEAM_BATTING_HR) %>% 
    mutate(TEAM_BATTING_AB = TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BATTING_HBP + TEAM_BATTING_SO) %>%
    mutate(TEAM_BATTING_AVG = TEAM_BATTING_H/TEAM_BATTING_AB) %>%
    mutate(TEAM_BATTING_OBP = (TEAM_BATTING_H + TEAM_BATTING_BB + TEAM_BATTING_HBP)/(TEAM_BATTING_AB + TEAM_BATTING_BB + TEAM_BATTING_HBP)) %>%
    mutate(TEAM_BATTING_SLG = (TEAM_BATTING_1B + 2*TEAM_BATTING_2B + 3*TEAM_BATTING_3B + 3*TEAM_BATTING_HR)/TEAM_BATTING_AB)
}

Given the presence of TEAM_BATTING_HBP in the computation of TEAM_BATTING_OBP, we need to impute the missing values, in this case with the mean of the variable.

raw_training_data <- read.csv("https://raw.githubusercontent.com/salma71/Data_621/master/HW_1/datasets/moneyball-training-data.csv", header = TRUE) %>% select(-INDEX)
data_train_mi <- raw_training_data
mean_hbp <- round(median(data_train_mi$TEAM_BATTING_HBP, na.rm = TRUE),0)
data_train_mi[is.na(data_train_mi[,"TEAM_BATTING_HBP"]), "TEAM_BATTING_HBP"] <- mean_hbp

data_train_mi <- add_advanced_bb_features(data_train_mi)
data_train_mi

data_train_mi %>%
  gather(variable, value, -c(TARGET_WINS:TEAM_BATTING_1B)) %>%
  ggplot(., aes(value, TARGET_WINS)) + 
  geom_point(fill = "pink", color="pink") + 
  geom_smooth(method = "lm", se = FALSE, color = "red") + 
  facet_wrap(~variable, scales ="free", nrow = 4) +
  labs(x = element_blank(), y = "Wins")

---

Models Building

Model_1.1 (without transformation)

Salma

# simple model
m1 <- lm(TARGET_WINS ~., data = data_train, na.action = na.omit)

summary(m1)

Call:
lm(formula = TARGET_WINS ~ ., data = data_train, na.action = na.omit)

Residuals:
    Min      1Q  Median      3Q     Max 
-49.112  -8.463   0.047   8.364  61.603 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)       6.9438573  5.8754934   1.182 0.237396    
TEAM_BATTING_H    0.0521644  0.0037624  13.865  < 2e-16 ***
TEAM_BATTING_2B  -0.0146400  0.0091385  -1.602 0.109290    
TEAM_BATTING_3B   0.0383492  0.0172017   2.229 0.025886 *  
TEAM_BATTING_HR   0.0768270  0.0281262   2.732 0.006354 ** 
TEAM_BATTING_BB   0.0013952  0.0049562   0.281 0.778354    
TEAM_BATTING_SO  -0.0019862  0.0033961  -0.585 0.558720    
TEAM_BASERUN_SB   0.0040365  0.0054119   0.746 0.455838    
TEAM_BASERUN_CS   0.0973314  0.0155751   6.249 4.92e-10 ***
TEAM_PITCHING_H  -0.0004574  0.0003778  -1.211 0.226086    
TEAM_PITCHING_HR  0.0030711  0.0247709   0.124 0.901341    
TEAM_PITCHING_BB  0.0108895  0.0032308   3.371 0.000763 ***
TEAM_PITCHING_SO -0.0023276  0.0022387  -1.040 0.298585    
TEAM_FIELDING_E  -0.0216762  0.0024116  -8.988  < 2e-16 ***
TEAM_FIELDING_DP -0.0961580  0.0139737  -6.881 7.65e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 12.95 on 2261 degrees of freedom
Multiple R-squared:  0.3284,    Adjusted R-squared:  0.3242 
F-statistic: 78.96 on 14 and 2261 DF,  p-value: < 2.2e-16

---

plot(m1)

---

Model_1.2 (Backward elimination)

m2 <- lm(TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR +TEAM_BATTING_3B +TEAM_BASERUN_CS+TEAM_FIELDING_E+ TEAM_FIELDING_DP, data = data_train)
summary(m2)

Call:
lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HR + 
    TEAM_BATTING_3B + TEAM_BASERUN_CS + TEAM_FIELDING_E + TEAM_FIELDING_DP, 
    data = data_train)

Residuals:
    Min      1Q  Median      3Q     Max 
-46.717  -8.663  -0.059   8.538  65.935 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)       4.473998   3.915403   1.143 0.253298    
TEAM_BATTING_H    0.051169   0.002467  20.741  < 2e-16 ***
TEAM_BATTING_HR   0.073968   0.007638   9.684  < 2e-16 ***
TEAM_BATTING_3B   0.061358   0.016375   3.747 0.000183 ***
TEAM_BASERUN_CS   0.107941   0.012081   8.935  < 2e-16 ***
TEAM_FIELDING_E  -0.023632   0.001584 -14.924  < 2e-16 ***
TEAM_FIELDING_DP -0.078465   0.013791  -5.690 1.44e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 13.06 on 2269 degrees of freedom
Multiple R-squared:  0.3144,    Adjusted R-squared:  0.3126 
F-statistic: 173.4 on 6 and 2269 DF,  p-value: < 2.2e-16

---

plot(m2)

---

Model_1.3 (polynomial regression)

full_formula <- "TARGET_WINS ~ TEAM_BATTING_2B + TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO + TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP + I(TEAM_BATTING_2B^2) + I(TEAM_BATTING_3B^2) + I(TEAM_BATTING_HR^2) + I(TEAM_BATTING_BB^2) + I(TEAM_BATTING_SO^2) + I(TEAM_BASERUN_SB^2) + I(TEAM_BASERUN_CS^2) + I(TEAM_PITCHING_H^2) + I(TEAM_PITCHING_HR^2) + I(TEAM_PITCHING_BB^2) + I(TEAM_PITCHING_SO^2) + I(TEAM_FIELDING_E^2) + I(TEAM_FIELDING_DP^2)  + I(TEAM_BATTING_2B^3) + I(TEAM_BATTING_3B^3) + I(TEAM_BATTING_HR^3) + I(TEAM_BATTING_BB^3) + I(TEAM_BATTING_SO^3) + I(TEAM_BASERUN_SB^3) + I(TEAM_BASERUN_CS^3) + I(TEAM_PITCHING_H^3) + I(TEAM_PITCHING_HR^3) + I(TEAM_PITCHING_BB^3) + I(TEAM_PITCHING_SO^3) + I(TEAM_FIELDING_E^3) + I(TEAM_FIELDING_DP^3)  + I(TEAM_BATTING_2B^4) + I(TEAM_BATTING_3B^4) + I(TEAM_BATTING_HR^4) + I(TEAM_BATTING_BB^4) + I(TEAM_BATTING_SO^4) + I(TEAM_BASERUN_SB^4) + I(TEAM_BASERUN_CS^4) + I(TEAM_PITCHING_H^4) + I(TEAM_PITCHING_HR^4) + I(TEAM_PITCHING_BB^4) + I(TEAM_PITCHING_SO^4) + I(TEAM_FIELDING_E^4) + I(TEAM_FIELDING_DP^4) "

m3 <- lm(full_formula, data_train)
step_back <- MASS::stepAIC(m3, direction="backward", trace = F)
poly_call <- summary(step_back)$call
step_back <- lm(poly_call[2], data_train)
summary(step_back)

Call:
lm(formula = poly_call[2], data = data_train)

Residuals:
    Min      1Q  Median      3Q     Max 
-54.736  -7.452   0.017   7.344  61.619 

Coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)           -1.055e+01  5.511e+01  -0.191 0.848198    
TEAM_BATTING_2B        1.519e+00  1.809e-01   8.396  < 2e-16 ***
TEAM_BATTING_3B        3.764e-01  1.871e-01   2.012 0.044312 *  
TEAM_BATTING_BB        7.492e-01  9.617e-02   7.790 1.02e-14 ***
TEAM_BASERUN_SB        4.464e-02  1.191e-02   3.748 0.000182 ***
TEAM_PITCHING_H        4.189e-02  3.584e-03  11.689  < 2e-16 ***
TEAM_PITCHING_BB      -6.975e-02  1.195e-02  -5.836 6.14e-09 ***
TEAM_PITCHING_SO      -2.545e-02  4.404e-03  -5.779 8.57e-09 ***
TEAM_FIELDING_E       -2.289e-01  2.252e-02 -10.166  < 2e-16 ***
TEAM_FIELDING_DP      -3.639e+00  1.629e+00  -2.233 0.025639 *  
I(TEAM_BATTING_2B^2)  -5.951e-03  7.210e-04  -8.254 2.58e-16 ***
I(TEAM_BATTING_3B^2)  -5.323e-03  3.696e-03  -1.440 0.149948    
I(TEAM_BATTING_HR^2)  -2.369e-03  9.823e-04  -2.412 0.015967 *  
I(TEAM_BATTING_BB^2)  -2.499e-03  3.194e-04  -7.825 7.78e-15 ***
I(TEAM_BATTING_SO^2)   3.270e-05  8.862e-06   3.690 0.000229 ***
I(TEAM_BASERUN_CS^2)  -1.157e-03  8.030e-04  -1.441 0.149837    
I(TEAM_PITCHING_H^2)  -4.327e-06  4.723e-07  -9.161  < 2e-16 ***
I(TEAM_PITCHING_HR^2)  3.015e-03  7.715e-04   3.909 9.56e-05 ***
I(TEAM_PITCHING_SO^2)  2.245e-06  1.159e-06   1.937 0.052925 .  
I(TEAM_FIELDING_E^2)   3.929e-04  5.385e-05   7.296 4.10e-13 ***
I(TEAM_FIELDING_DP^2)  3.887e-02  1.806e-02   2.153 0.031455 *  
I(TEAM_BATTING_2B^3)   7.573e-06  9.411e-07   8.047 1.36e-15 ***
I(TEAM_BATTING_3B^3)   4.442e-05  2.803e-05   1.585 0.113173    
I(TEAM_BATTING_HR^3)   1.575e-05  7.177e-06   2.194 0.028302 *  
I(TEAM_BATTING_BB^3)   3.690e-06  4.879e-07   7.564 5.68e-14 ***
I(TEAM_BATTING_SO^3)  -2.345e-08  5.875e-09  -3.992 6.76e-05 ***
I(TEAM_BASERUN_SB^3)  -2.227e-07  1.443e-07  -1.543 0.122933    
I(TEAM_BASERUN_CS^3)   2.134e-05  9.901e-06   2.156 0.031216 *  
I(TEAM_PITCHING_H^3)   1.688e-10  2.696e-11   6.262 4.54e-10 ***
I(TEAM_PITCHING_HR^3) -1.683e-05  4.989e-06  -3.374 0.000753 ***
I(TEAM_PITCHING_BB^3)  1.061e-08  2.997e-09   3.540 0.000408 ***
I(TEAM_FIELDING_E^3)  -3.019e-07  4.806e-08  -6.281 4.02e-10 ***
I(TEAM_FIELDING_DP^3) -1.854e-04  8.639e-05  -2.146 0.031983 *  
I(TEAM_BATTING_3B^4)  -1.265e-07  6.979e-08  -1.813 0.069995 .  
I(TEAM_BATTING_HR^4)  -2.933e-08  1.495e-08  -1.962 0.049907 *  
I(TEAM_BATTING_BB^4)  -1.874e-09  2.652e-10  -7.064 2.16e-12 ***
I(TEAM_BASERUN_SB^4)   3.276e-10  2.064e-10   1.587 0.112665    
I(TEAM_BASERUN_CS^4)  -6.719e-08  3.306e-08  -2.032 0.042243 *  
I(TEAM_PITCHING_H^4)  -2.130e-15  4.961e-16  -4.295 1.82e-05 ***
I(TEAM_PITCHING_HR^4)  2.649e-08  8.868e-09   2.988 0.002842 ** 
I(TEAM_PITCHING_BB^4) -1.522e-12  6.952e-13  -2.189 0.028685 *  
I(TEAM_FIELDING_E^4)   7.607e-11  1.416e-11   5.371 8.66e-08 ***
I(TEAM_FIELDING_DP^4)  3.253e-07  1.511e-07   2.153 0.031439 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 11.95 on 2233 degrees of freedom
Multiple R-squared:  0.4351,    Adjusted R-squared:  0.4245 
F-statistic: 40.95 on 42 and 2233 DF,  p-value: < 2.2e-16

---

plot(step_back)

---

Model_2.1 (using transformation)

Using Transformation - Dhairav

#data with all transformed variables
m4_data <- select(data_transformed,-TEAM_BASERUN_CS, -TEAM_BASERUN_SB, -TEAM_BATTING_3B, -TEAM_BATTING_BB, -TEAM_BATTING_H, -TEAM_FIELDING_E, -TEAM_PITCHING_BB, -TEAM_PITCHING_H, -TEAM_PITCHING_SO, -TEAM_BATTING_HBP, -TEAM_BASERUN_CS, -TEAM_BATTING_1B)

m4 <- lm(TARGET_WINS ~., data = m4_data)
summary(m4)

Call:
lm(formula = TARGET_WINS ~ ., data = m4_data)

Residuals:
    Min      1Q  Median      3Q     Max 
-33.994  -6.794   0.131   6.903  31.095 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)            -5.713e+04  3.049e+04  -1.873 0.061213 .  
TEAM_BATTING_2B        -5.845e-02  2.991e-02  -1.954 0.050836 .  
TEAM_BATTING_HR         8.265e-02  7.497e-02   1.102 0.270442    
TEAM_BATTING_SO        -3.289e-02  9.775e-03  -3.365 0.000786 ***
TEAM_PITCHING_HR        2.690e-02  6.422e-02   0.419 0.675433    
TEAM_FIELDING_DP       -1.084e-01  1.343e-02  -8.069 1.46e-15 ***
TEAM_BASERUN_CS_tform   9.830e-01  1.067e+00   0.921 0.356943    
TEAM_BASERUN_SB_tform   3.803e+00  8.515e-01   4.466 8.57e-06 ***
TEAM_BATTING_3B_tform   6.310e+00  1.456e+00   4.335 1.56e-05 ***
TEAM_BATTING_BB_tform   5.846e-05  2.713e-05   2.155 0.031329 *  
TEAM_BATTING_H_tform    8.573e+04  7.161e+04   1.197 0.231415    
TEAM_BATTING_1B_tform   1.446e+04  3.134e+04   0.462 0.644473    
TEAM_FIELDING_E_tform  -1.168e+03  9.249e+01 -12.630  < 2e-16 ***
TEAM_PITCHING_BB_tform  4.444e+00  8.335e+00   0.533 0.594005    
TEAM_PITCHING_H_tform  -4.167e+04  2.492e+04  -1.672 0.094777 .  
TEAM_PITCHING_SO_tform  8.444e+00  7.733e+00   1.092 0.275075    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 9.828 on 1470 degrees of freedom
  (790 observations deleted due to missingness)
Multiple R-squared:  0.4066,    Adjusted R-squared:  0.4005 
F-statistic: 67.14 on 15 and 1470 DF,  p-value: < 2.2e-16

---

Model_2.2 (Backward elimination)

Using Stepwise backward elimination

m4a <- update(m4, . ~ . -TEAM_PITCHING_HR)
# summary(m4a)
m4b <- update(m4a, . ~ . -TEAM_BATTING_1B_tform)
# summary(m4b)
m4c <- update(m4b, . ~ . -TEAM_PITCHING_BB_tform)
# summary(m4c)
m4d <- update(m4c, . ~ . -TEAM_PITCHING_SO_tform)
# summary(m4d)
m4e <- update(m4d, . ~ . -TEAM_PITCHING_H_tform)
summary(m4e)

Call:
lm(formula = TARGET_WINS ~ TEAM_BATTING_2B + TEAM_BATTING_HR + 
    TEAM_BATTING_SO + TEAM_FIELDING_DP + TEAM_BASERUN_CS_tform + 
    TEAM_BASERUN_SB_tform + TEAM_BATTING_3B_tform + TEAM_BATTING_BB_tform + 
    TEAM_BATTING_H_tform + TEAM_FIELDING_E_tform, data = m4_data)

Residuals:
    Min      1Q  Median      3Q     Max 
-33.525  -7.082   0.271   6.968  31.845 

Coefficients:
                        Estimate Std. Error t value Pr(>|t|)    
(Intercept)           -6.994e+04  9.958e+03  -7.024 3.29e-12 ***
TEAM_BATTING_2B       -7.585e-02  9.613e-03  -7.890 5.85e-15 ***
TEAM_BATTING_HR        9.978e-02  9.736e-03  10.249  < 2e-16 ***
TEAM_BATTING_SO       -2.190e-02  2.503e-03  -8.750  < 2e-16 ***
TEAM_FIELDING_DP      -1.084e-01  1.340e-02  -8.086 1.27e-15 ***
TEAM_BASERUN_CS_tform  7.508e-01  1.061e+00   0.707    0.479    
TEAM_BASERUN_SB_tform  4.026e+00  8.372e-01   4.809 1.67e-06 ***
TEAM_BATTING_3B_tform  5.518e+00  9.491e-01   5.814 7.45e-09 ***
TEAM_BATTING_BB_tform  7.186e-05  6.130e-06  11.723  < 2e-16 ***
TEAM_BATTING_H_tform   7.145e+04  9.968e+03   7.168 1.20e-12 ***
TEAM_FIELDING_E_tform -1.190e+03  9.144e+01 -13.015  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 9.835 on 1475 degrees of freedom
  (790 observations deleted due to missingness)
Multiple R-squared:  0.4037,    Adjusted R-squared:  0.3997 
F-statistic: 99.88 on 10 and 1475 DF,  p-value: < 2.2e-16

plot(m4e)

---

Model_3.1 (Feature Engineering)

Mael

mi_m1 <- lm(TARGET_WINS ~ TEAM_BATTING_AB + TEAM_BATTING_AVG + TEAM_BATTING_OBP + TEAM_BATTING_SLG, data = data_train_mi)
summary(mi_m1)

Call:
lm(formula = TARGET_WINS ~ TEAM_BATTING_AB + TEAM_BATTING_AVG + 
    TEAM_BATTING_OBP + TEAM_BATTING_SLG, data = data_train_mi)

Residuals:
    Min      1Q  Median      3Q     Max 
-64.096  -8.716   0.584   9.161  52.118 

Coefficients:
                   Estimate Std. Error t value Pr(>|t|)    
(Intercept)      -9.015e+01  8.608e+00 -10.473  < 2e-16 ***
TEAM_BATTING_AB   2.508e-02  1.923e-03  13.038  < 2e-16 ***
TEAM_BATTING_AVG -2.207e+02  3.144e+01  -7.019 2.97e-12 ***
TEAM_BATTING_OBP  2.671e+02  3.149e+01   8.481  < 2e-16 ***
TEAM_BATTING_SLG  7.515e+01  1.213e+01   6.195 6.93e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 13.79 on 2169 degrees of freedom
  (102 observations deleted due to missingness)
Multiple R-squared:  0.2173,    Adjusted R-squared:  0.2158 
F-statistic: 150.5 on 4 and 2169 DF,  p-value: < 2.2e-16

The standardized residual plots show quite a few points outside the -2,2 range, which might justify removing those observations.

Residuals vs Fitted: while the line is not quite horizontal, the constant variance assumption seems met Normal Q-Q plot: normality assumption is met Root(Squared Residuals) vs Fitted Values: Residuals vs Leverage: a few points have standardized residuals outside the (-2,2) ranhe which might justify removing those observations.

plot(mi_m1)

MI: tyring out predictions on this model

# data_test_mi <- add_advanced_bb_features(data_test_mi)
# pred.w.plim <- predict(mi_m1, data_test_mi, interval = "prediction")
# pred.w.clim <- predict(mi_m1, data_test_mi, interval = "confidence")

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Model selection